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Peter Monk - One of the best experts on this subject based on the ideXlab platform.
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Improvements for the ultra weak Variational Formulation
International Journal for Numerical Methods in Engineering, 2013Co-Authors: Teemu Luostari, Tomi Huttunen, Peter MonkAbstract:SUMMARY In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak Variational Formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two-dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.
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Error estimates for the Ultra Weak Variational Formulation of the Helmholtz equation
ESAIM: Mathematical Modelling and Numerical Analysis, 2008Co-Authors: Annalisa Buffa, Peter MonkAbstract:The Ultra Weak Variational Formulation (UWVF) of the Helmholtz equation provides a Variational framework suitable for discretization using plane wave solutions of an appropriate adjoint equation. Currently convergence of the method is only proved on the boundary of the domain. However substantial computational evidence exists showing that the method also converges throughout the domain of the Helmholtz equation. In this paper we exploit the fact that the UWVF is essentially an upwind discontinuous Galerkin method to prove convergence of the solution in the special case where there is no absorbing medium present. We also provide some other estimates in the case when absorption is present, and give some simple numerical results to test the estimates. We expect that similar techniques can be used to prove error estimates for the UWVF applied to Maxwell's equations and elasticity.
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Solving Maxwell's equations using the ultra weak Variational Formulation
Journal of Computational Physics, 2007Co-Authors: Tomi Huttunen, M. Malinen, Peter MonkAbstract:We investigate the ultra weak Variational Formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an electromagnetic UWVF solver. In particular, we propose a method to avoid the conditioning problems that are known to hamper the use of the UWVF for problems in general geometries and inhomogeneous media. In addition, we show how to implement the PML in the UWVF to accurately approximate physically unbounded problems and discuss the parallelization of the UWVF. Three-dimensional numerical simulations are used to examine the feasibility of the UWVF for simulating wave propagation in inhomogeneous media and scattering from complex structures.
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The Ultra-Weak Variational Formulation for Elastic Wave Problems
SIAM Journal on Scientific Computing, 2004Co-Authors: Tomi Huttunen, Peter Monk, F. Collino, Jari P. KaipioAbstract:The ultra-weak Variational Formulation has been used effectively to solve time-harmonic acoustic and electromagnetic wave propagation in inhomogeneous media. We develop the ultra-weak Variational Formulation for elastic wave propagation in two space dimensions. In order to improve the accuracy and stability of the method, we find it necessary to approximate the S- and P-wave components of the solution in a balanced way. Some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements.
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Computational aspects of the ultra-weak Variational Formulation
Journal of Computational Physics, 2002Co-Authors: Tomi Huttunen, Peter Monk, Jari P. KaipioAbstract:The ultra-weak Variational Formulation (UWVF) approach has been proposed as an effective method for solving Helmholtz problems with high wave numbers. However, for coarse meshes the method can suffer from instability. In this paper we consider computational aspects of the ultra-weak Variational Formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme and we compare iterative solvers for the resulting linear system. Computations for the acoustic transmission problem in 2D show that the new approach enables Helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers.
Tomi Huttunen - One of the best experts on this subject based on the ideXlab platform.
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Improvements for the ultra weak Variational Formulation
International Journal for Numerical Methods in Engineering, 2013Co-Authors: Teemu Luostari, Tomi Huttunen, Peter MonkAbstract:SUMMARY In this paper, we investigate strategies to improve the accuracy and efficiency of the ultra weak Variational Formulation (UWVF) of the Helmholtz equation. The UWVF is a Trefftz type, nonpolynomial method using basis functions derived from solutions of the adjoint Helmholtz equation. We shall consider three choices of basis function: propagating plane waves (original choice), Bessel basis functions, and evanescent wave basis functions. Traditionally, two-dimensional triangular elements are used to discretize the computational domain. However, the element shapes affect the conditioning of the UWVF. Hence, we investigate the use of different element shapes aiming to lower the condition number and number of degrees of freedom. Our results include the first tests of a plane wave method on meshes of mixed element types. In many modeling problems, evanescent waves occur naturally and are challenging to model. Therefore, we introduce evanescent wave basis functions for the first time in the UWVF to tackle rapidly decaying wave modes. The advantages of an evanescent wave basis are verified by numerical simulations on domains including curved interfaces.Copyright © 2013 John Wiley & Sons, Ltd.
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Solving Maxwell's equations using the ultra weak Variational Formulation
Journal of Computational Physics, 2007Co-Authors: Tomi Huttunen, M. Malinen, Peter MonkAbstract:We investigate the ultra weak Variational Formulation for simulating time-harmonic Maxwell problems. This study has two main goals. First, we introduce a novel derivation of the UWVF method which shows that the UWVF is an unusual version of the standard upwind discontinuous Galerkin (DG) method with a special choice of basis functions. Second, we discuss the practical implementation of an electromagnetic UWVF solver. In particular, we propose a method to avoid the conditioning problems that are known to hamper the use of the UWVF for problems in general geometries and inhomogeneous media. In addition, we show how to implement the PML in the UWVF to accurately approximate physically unbounded problems and discuss the parallelization of the UWVF. Three-dimensional numerical simulations are used to examine the feasibility of the UWVF for simulating wave propagation in inhomogeneous media and scattering from complex structures.
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The Ultra-Weak Variational Formulation for Elastic Wave Problems
SIAM Journal on Scientific Computing, 2004Co-Authors: Tomi Huttunen, Peter Monk, F. Collino, Jari P. KaipioAbstract:The ultra-weak Variational Formulation has been used effectively to solve time-harmonic acoustic and electromagnetic wave propagation in inhomogeneous media. We develop the ultra-weak Variational Formulation for elastic wave propagation in two space dimensions. In order to improve the accuracy and stability of the method, we find it necessary to approximate the S- and P-wave components of the solution in a balanced way. Some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements.
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Computational aspects of the ultra-weak Variational Formulation
Journal of Computational Physics, 2002Co-Authors: Tomi Huttunen, Peter Monk, Jari P. KaipioAbstract:The ultra-weak Variational Formulation (UWVF) approach has been proposed as an effective method for solving Helmholtz problems with high wave numbers. However, for coarse meshes the method can suffer from instability. In this paper we consider computational aspects of the ultra-weak Variational Formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme and we compare iterative solvers for the resulting linear system. Computations for the acoustic transmission problem in 2D show that the new approach enables Helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers.
Jari P. Kaipio - One of the best experts on this subject based on the ideXlab platform.
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The Ultra-Weak Variational Formulation for Elastic Wave Problems
SIAM Journal on Scientific Computing, 2004Co-Authors: Tomi Huttunen, Peter Monk, F. Collino, Jari P. KaipioAbstract:The ultra-weak Variational Formulation has been used effectively to solve time-harmonic acoustic and electromagnetic wave propagation in inhomogeneous media. We develop the ultra-weak Variational Formulation for elastic wave propagation in two space dimensions. In order to improve the accuracy and stability of the method, we find it necessary to approximate the S- and P-wave components of the solution in a balanced way. Some preliminary analysis is provided and numerical evidence is presented for the efficiency of the scheme in comparison to piecewise linear finite elements.
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Computational aspects of the ultra-weak Variational Formulation
Journal of Computational Physics, 2002Co-Authors: Tomi Huttunen, Peter Monk, Jari P. KaipioAbstract:The ultra-weak Variational Formulation (UWVF) approach has been proposed as an effective method for solving Helmholtz problems with high wave numbers. However, for coarse meshes the method can suffer from instability. In this paper we consider computational aspects of the ultra-weak Variational Formulation for the inhomogeneous Helmholtz problem. We introduce a method to improve the UWVF scheme and we compare iterative solvers for the resulting linear system. Computations for the acoustic transmission problem in 2D show that the new approach enables Helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers.
Francois Gaybalmaz - One of the best experts on this subject based on the ideXlab platform.
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single and double generator bracket Formulations of multicomponent fluids with irreversible processes
Journal of Physics A, 2020Co-Authors: Christopher Eldred, Francois GaybalmazAbstract:The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known Variational Formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian Formulation that involves a Poisson bracket structure. However, real flows also include irreversible processes, such as viscous dissipation, heat conduction, diffusion and phase changes. Recent work has demonstrated that the Variational Formulation can be systematically extended to include irreversible processes and nonequilibrium thermodynamics, through the new concept of thermodynamic displacement. Irreversible processes have also been incorporated into the bracket structure through the addition of a dissipation bracket. This gives what are known as the single and double generator bracket Formulations, which are the natural generalizations of the Hamiltonian Formulation to include irreversible dynamics. Unlike the Variational formalism, most of these bracket formalisms do not follow from a systematic construction and have often been derived via a case by case approach, with slightly different axioms used in different situations. In this paper, we show that the Variational Formulation yields a constructive and systematic way to derive from a unified perspective these bracket Formulations for fully compressible, multicomponent, multiphase fluids with a single temperature and velocity. In the case of a linear relation between the thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC brackets are recovered. Many previous results in the literature, typically obtained via heuristic approaches, are demonstrated to be special cases of this general Formulation.
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a free energy lagrangian Variational Formulation of the navier stokes fourier system
International Journal of Geometric Methods in Modern Physics, 2019Co-Authors: Francois Gaybalmaz, Hiroaki YoshimuraAbstract:We present a Variational Formulation for the Navier–Stokes–Fourier system based on a free energy Lagrangian. This Formulation is a systematic infinite-dimensional extension of the Variational appro...
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single and double generator bracket Formulations of geophysical fluids with irreversible processes
arXiv: Classical Physics, 2018Co-Authors: Christopher Eldred, Francois GaybalmazAbstract:The equations of reversible (inviscid, adiabatic) fluid dynamics have a well-known Variational Formulation based on Hamilton's principle and the Lagrangian, to which is associated a Hamiltonian Formulation that involves a Poisson bracket structure. These Variational and bracket structures underlie many of the most basic principles that we know about geophysical fluid flows, such as conservation laws. However, real geophysical flows also include irreversible processes, such as viscous dissipation, heat conduction, diffusion and phase changes. Recent work has demonstrated that the Variational Formulation can be extended to include irreversible processes and non-equilibrium thermodynamics, through the new concept of thermodynamic displacement. By design, and in accordance with fundamental physical principles, the resulting equations automatically satisfy the first and second law of thermodynamics. Irreversible processes can also be incorporated into the bracket structure through the addition of a dissipation bracket. This gives what are known as the single and double generator bracket Formulations, which are the natural generalizations of the Hamiltonian Formulation to include irreversible dynamics. Here the Variational Formulation for irreversible processes is shown to underlie these bracket Formulations for fully compressible, multicomponent, multiphase geophysical fluids with a single temperature and velocity. Many previous results in the literature are demonstrated to be special cases of this approach. Finally, some limitations of the current approach (especially with regards to precipitation and nonlocal processes such as convection) are discussed, and future directions of research to overcome them are outlined.
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a free energy lagrangian Variational Formulation of the navier stokes fourier system
arXiv: Mathematical Physics, 2017Co-Authors: Francois Gaybalmaz, Hiroaki YoshimuraAbstract:We present a Variational Formulation for the Navier-Stokes-Fourier system based on a free energy Lagrangian. This Formulation is a systematic infinite dimensional extension of the Variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian Variational Formulation using the internal energy developed in \cite{GBYo2016b} as one employs temperature, rather than entropy, as an independent variable. The Variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The Variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier-Stokes-Fourier system on Riemannian manifolds.
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a lagrangian Variational Formulation for nonequilibrium thermodynamics part i discrete systems
Journal of Geometry and Physics, 2017Co-Authors: Francois Gaybalmaz, Hiroaki YoshimuraAbstract:Abstract In this paper, we present a Lagrangian Variational Formulation for nonequilibrium thermodynamics. This Formulation is an extension of Hamilton’s principle of classical mechanics that allows the inclusion of irreversible phenomena. The irreversibility is encoded into a nonlinear phenomenological constraint given by the expression of the entropy production associated to all the irreversible processes involved. From a mathematical point of view, our Variational Formulation may be regarded as a generalization to nonequilibrium thermodynamics of the Lagrange–d’Alembert principle used in nonlinear nonholonomic mechanics, where the conventional Lagrange–d’Alembert principle cannot be applied since the nonlinear phenomenological constraint and its associated Variational constraint must be treated separately. In our approach, to deal with the nonlinear nonholonomic constraint, we introduce a variable called the thermodynamic displacement associated to each irreversible process. This allows us to systematically define the corresponding Variational constraint. In Part I, our Variational theory is illustrated with various examples of discrete systems such as mechanical systems with friction, matter transfer, electric circuits, chemical reactions, and diffusion across membranes. In Part II of the present paper, we will extend our Variational Formulation of discrete systems to the case of continuum systems.
M Profit - One of the best experts on this subject based on the ideXlab platform.
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Variational Formulation for the smooth particle hydrodynamics sph simulation of fluid and solid problems
Computer Methods in Applied Mechanics and Engineering, 2004Co-Authors: Javier Bonet, Sivakumar Kulasegaram, M X Rodriguezpaz, M ProfitAbstract:The paper describes the Variational Formulation of smooth particle hydrodynamics for both fluids and solids applications. The resulting equations treat the continuum as a Hamiltonian system of particles where the constitutive equation of the continuum is represented via an internal energy term. For solids this internal energy is derived from the deformation gradient of the mapping in terms of a hyperelastic strain energy function. In the case of fluids, the internal energy term is a function of the density. Once the internal energy terms are established the equations of motion are developed as equations of Lagrange, where the Lagrangian coordinates are the current positions of the particles. Since the energy terms are independent of rigid body rotations and translations, this Formulation ensures the preservation of physical constants of the motion such as linear and angular momentum.
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a Variational Formulation based contact algorithm for rigid boundaries in two dimensional sph applications
Computational Mechanics, 2004Co-Authors: Sivakumar Kulasegaram, Javier Bonet, R W Lewis, M ProfitAbstract:Smooth particle Hydrodynamics (SPH) is one of the most effective meshless techniques used in computational mechanics. SPH approximations are simple and allow greater flexibility in various engineering applications. However, modelling of particle-boundary interactions in SPH computations has always been considered an aspect that requires further research. A number of techniques have been developed to model particle-boundary interactions in SPH and allied methods. In this paper, an innovative approach is introduced to handle the contact between Lagrangian SPH particles and rigid solid boundaries. The Formulation of boundary contact forces are derived based on a Variational Formulation, thus directly ensuring the conservativeness of the governing equations. In addition, the new elegant boundary contact force terms maintain the simplicity of the SPH governing equations.
